SATHEE: Probability 3 Question 34

Probability 3 Question 34

34. If $A$ and $B$ are two independent events, prove that $P (A \cup B) \cdot P (A^{'} \cap B^{'}) \leq P (C)$ , where $C$ is an event defined that exactly one of $A$ and $B$ occurs. (2004, 2M)

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Solution:

Here, $P (A \cup B) \cdot P (A^{'} \cap B^{'})$

$\Rightarrow P (A) + P (B) - P (A \cap B) P (A^{'}) \cdot P (B^{'})$

[since $A, B$ are independent, so $A^{'}, B^{'}$ are independent]

$∴ P (A \cup B) \cdot P (A^{'} \cap B^{'}) \leq P (A) + P (B) \cdot P (A^{'}) \cdot P (B)$

$= P (A) \cdot P (A^{'}) \cdot P (B) + P (B) \cdot P (A^{'}) \cdot P (B)$

$\leq P (A) \cdot P (B) + P (B) \cdot P (A^{'})$

$[∵ P (A^{'}) \leq 1 and P (B^{'}) \leq 1]$

$\Rightarrow P (A \cup B) \cdot P (A^{'} \cap B^{'}) \leq P (A) \cdot P (B^{'}) + P (B) \cdot P (A^{'})$

$\Rightarrow P (A \cup B) \cdot P (A^{'} \cap B^{'}) \leq P (C)$

$[∵ P (C) = P (A) \cdot P (B^{'}) + P (B) \cdot P (A^{'})]$

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Probability 3 Question 34

34. If A and B are two independent events, prove that P(A∪B)⋅P(A′∩B′)≤P(C), where C is an event defined that exactly one of A and B occurs. (2004, 2M)

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34. If $A$ and $B$ are two independent events, prove that $P (A \cup B) \cdot P (A^{'} \cap B^{'}) \leq P (C)$ , where $C$ is an event defined that exactly one of $A$ and $B$ occurs. (2004, 2M)